In this paper we derive the analogue of the Lebesque-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on Zp. 2010 Mathematics Subject Classification : 11S80, 48B22, 28B99

The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As applications, among other results, versions of the Banach-Steinhaus theorem and the Nikodym boundednes...

These notes give various references to the literature that I will not, for reasons of brevity, give during the lectures themselves. 1. Gaussian Measures A comprehensive reference for Gaussian measures is [Bog98]. This book works in the generality of locally convex topological spaces, and has material on Gaussian measure on Banach and Hilbert spaces as particular cases. A similar book, focussed ...

Conditional expectation is one of the most useful tools of probability. The Radon-Nikodym theorem enables us to construct conditional expectations. Definition 1. A measure µ on (Ω, F) is σ-finite iff there is A n ∈ F satisfying A n ↑ Ω where µ(A n) < ∞ for all n. Definition 2. Let µ, ν be measures on (Ω, F). Then ν is absolutely continuous with respect to µ (ν << µ) iff for all A ∈ F, µ(A) = 0 ...